# Algebraic technique · Grade 10 · 4 units

> Algebraic technique, Grade 10 · 4 units: Factorising, Powers, Roots and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Algebraic technique (Grade 10 · 4 units) you practise 5 skills: Factorising · Adding and subtracting algebraic fractions · Simplifying, multiplying and dividing algebraic fractions · Powers · Roots.

Factorising turns a sum into a product. Start with a common factor, then the special products, and factorise a trinomial through its roots: when the leading coefficient is not one, find two numbers whose product is the leading coefficient times the constant term and whose sum is the middle coefficient.

To add algebraic fractions, bring them to a common denominator: factorise each denominator and take the smallest product that every denominator divides. Before you simplify, factorise the numerator and the denominator, and cancel only factors, never terms. For every algebraic fraction, remember which values are not allowed: the ones that make the denominator zero.

A negative power is one over the positive power, and any number other than zero to the power of zero is one. The square root of a product is the product of the square roots, which is how you take a factor out of a root: look for a perfect square inside it.

**Tip:** After each factorisation, multiply back in your head: if you get the expression you started with, the factorisation is right.

## What you learn

### Factorising

- Example: Factorise fully: 6x + 12
- Answer: 6(x + 2)
- Solution: The common factor: 6 What is left in the bracket: x + 2 Check by expanding back to the original expression: 6(x + 2) = 6x + 12

### Adding and subtracting algebraic fractions

- Example: Subtract and simplify: (5x − 5) ÷ (x + 3) − (3x + 5) ÷ (x + 3)
- Options: (2x − 10) ÷ (2x + 6) · (2x − 10) ÷ (x + 3)² · 2x ÷ (x + 3) · (2x − 10) ÷ (x + 3)
- Answer: (2x − 10) ÷ (x + 3)
- Solution: The common denominator: x + 3 . Scale up and combine the numerators: (5x − 5) ÷ (x + 3) − (3x + 5) ÷ (x + 3) = (5x − 5 − (3x + 5)) ÷ (x + 3) = (2x − 10) ÷ (x + 3)

### Simplifying, multiplying and dividing algebraic fractions

- Example: Simplify: 12x² ÷ 2x
- Answer: 6x
- Solution: Factorise the numerator: 2x × 6x ÷ 2x Cancel: 6x

### Powers

- Example: Write as a single power: 10² × 10⁴
- Answer: 6
- Solution: Add the indices: 2 + 4 = 6 . So 10² × 10⁴ = 10⁶

### Roots

- Example: Work out: √2 × √8
- Answer: 4
- Solution: √2 × √8 = √(2 × 8) = √16 = 4

## Not on the site yet

- Cubic special products
- Factorising four terms by grouping

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

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